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Time Dependent Perturbationdependent perturbation from Galilean Transformationtransformation

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Time Dependent Perturbation from Galilean Transformation

I'm having a little trouble figuring out how to start this. The question is as follows. At time $t<0$ a hydrogen atom is in the rest frame $\mathfrak{R}$. Then at time $t=0$ the atom suddenly starts moving at a speed $V$ which is non-relativistic. It is now described in the rest frame $\mathfrak{R}'$ under the transformation $$t' = t, \, \, \, \vec{r}' = \vec{r}-\vec{V}t$$ The question asks to find the probability that at time $t>0$ in rest frame $\mathfrak{R}'$ that the electron in the hydrogen atom is still in it's ground state.

I know so far that this is a time dependent perturbation theory. I also know the probability can be computed by using $$|C_m^{(1)}(t)|^2$$ where $$C_m^{(1)}(t) = -\frac{i}{\hbar}\int_0^t dt e^{i\omega_{m,i}t}H'_{m,i}(t)$$ and $$\omega_{m,i}= \frac{E_m-E_i}{\hbar}, \, \, \, H'_{m,i} = \langle\psi_m^0|\hat{H}'(t)|\psi_i^0\rangle$$ What I'm not clear on is what $E_m$, $\psi_m^0$ and $\hat{H}'(t)$ should be. Would these just be the Galilean transformations of $E_i$, $\psi_i^0$ and $\hat{H}(t)$ for a hydrogen atom, or is there something more subtle I'm missing here?